document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=237368>Question 237368</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I had a question in my exam today and i couldnt answer it fully <BR>\n" );
document.write( "can you please help\r<BR>\n" );
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document.write( "Qeustion)a) <BR>\n" );
document.write( "find the set of values of 'k' for which the equation 'f(x)=0' has no real solutions.<BR>\n" );
document.write( "answer)<BR>\n" );
document.write( "f(x)=x^2-kx+9, where k is a constant<BR>\n" );
document.write( "b^2-4ac<0<BR>\n" );
document.write( "k^2-4*1*9=k^2-36<0<BR>\n" );
document.write( "k^2<36<BR>\n" );
document.write( "k<6<BR>\n" );
document.write( "-6<k<6<BR>\n" );
document.write( "(-6,6)<BR>\n" );
document.write( "that was fairly simple, however i  struggled on the next part<BR>\n" );
document.write( "Question)b) <BR>\n" );
document.write( "Given that k=4,<BR>\n" );
document.write( "express f(x) in the form (x-p)^2+q, where p and q are constants to be found.<BR>\n" );
document.write( "answer) <BR>\n" );
document.write( "Please help with this question because i really was angry when i couldnt complete it in my exam<BR>\n" );
document.write( "thank you for your cooporation.\r<BR>\n" );
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document.write( "by the way i used '^' to show power of<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #174571 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=174571\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> b) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-kx%2B9 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-kx%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-4x%2B9 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-4x%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=4 ALIGN=MIDDLE  ALT=\"k=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's complete the square.\r<BR>\n" );
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document.write( "To do so, take half of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28-4%29=-2 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28-4%29=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%5E2=%28-2%29%28-2%29=4 ALIGN=MIDDLE  ALT=\"%28-2%29%5E2=%28-2%29%28-2%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-4x%2Bhighlight%284-4%29%2B9 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-4x%2Bhighlight%284-4%29%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <font size=4><b>and</b></font> subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Make sure to place this after the \"x\" term. Notice how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-4=0 ALIGN=MIDDLE  ALT=\"4-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the expression is not changed.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2-4x%2B4%29-4%2B9 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2-4x%2B4%29-4%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group the first three terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x-2%29%5E2-4%2B9 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x-2%29%5E2-4%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4x%2B4 ALIGN=MIDDLE  ALT=\"x%5E2-4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x-2%29%5E2%2B5 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x-2%29%5E2%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So after completing the square, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4x%2B9 ALIGN=MIDDLE  ALT=\"x%5E2-4x%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > transforms to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2B5 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4x%2B9=%28x-2%29%5E2%2B5 ALIGN=MIDDLE  ALT=\"x%5E2-4x%2B9=%28x-2%29%5E2%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-4x%2B9 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-4x%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x-2%29%5E2%2B5 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x-2%29%5E2%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x-2%29%5E2%2B5 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x-2%29%5E2%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x-p%29%5E2%2Bq ALIGN=MIDDLE  ALT=\"f%28x%29=%28x-p%29%5E2%2Bq\"   DISABLED_style= \"max-width:90%; height:auto;\" > where p=2 and q=5. </SPAN>\n" );
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